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1 


1-0  di. 


h’IRST  LECTURE' 

Solution  of  equations  by  passing 
to  the  limit  from  Approximations.; 


The  problems  that  arise  in  algebra  are  of  a finite  character. 


These  meet  their  first  generalization  in  the  study  of  infinite  series 
in  which  the  notion  of  sum  is  generalized  from  the  finite  to  the 
infinite  case.;  We  meet  a similar  generalization  in  the  Riemann 
definition  of  integral.’  In  this  well*“known  case  an  interval  of 
integration^  from  to  Xn  j is  divided  into  ri  sub— interval  s^  the 
interval  from  Xj^  having  a lent^th  ,*2^.  The  function  value 

of  the  integrand  is  taken  at  some  point  in  the  interval^  and 

the  sum  formed 


^ 1,.  C,  n. 

The  limit  of  this  sum  if  it  exists^,  for  any  method  of  subdivision 
into  n parts  as  n increases  indefinitely^  the  length  of  the  maximum 
'hj  decreasing  indefinitely  with  1 ^ n,  is  the  Riemann  integral.' 

Likewise  in  Jefinin^  a derivative  we  pass  from  a finite  case 
to  an  infinite  case.;  Usually  the  derivative  is  the  inverse  of  the 
integral  defined  above^  but  it  was  pointed  out  by  Dini  that  there 
were  functions  which  had  derivatives  but  such  that  the  derivative 
was  not  inte^rable  in  the  Riemann  sense,'  On  the  basis  of  his  own 
and  also  investigations  of  Sorely  Lebes^ue  «jave  a definition  of 
intCiJral  which  was  much  1 reader  in  extent^  and  took  care  of  some  of 
these  exceptional  cases  ; 

The  definition  of  derivative  is  usually  ^iven  in  the  form 

F( X* h)-F{ x) 


This  can  be  determined  approximately  by  taking  \ as  small  as  possible 


4U 


^iifilaa^o  %<,i  isnoiifEupa  lo  noiJuio2 
S«floli£(£iixo'iaqA  atot^  iiiail 


' 3 'i  ‘'\ 


H, 


*5^iffx3  £ ![o  97K  ut^ajda  ni  &8ti&  ^utrl^  eojaldoiq  S 


:Wji 


.SaiTae  «-‘i(.Mni  to  ,iuJ8  ni  noxias  ileisnss  Xanii  nifux,,, 


9.-IJ  oJ  sXini^  sdj  „otl  tosilaianss  si  lua  }o  notion  erfj  iioHw  n 

■■•j  ■■  ' li  ^ 

nna^nsifl  srIJ  ni  noiissiXe-isnss  isliaia  a Jssn  sW  i.saat  aiinilni 


RV, 


io  Xavisini  na  asao  n»onH-XX,»  airii  nl  ;.Xeis,s.ini  io  poilinilal 
arii  ..sla.naini-dua  „ oini  iabivit^si  oi  ,x  »on1  .noixan^ain) 
»tlav  noixonai  aPT  .jP  rfx.nnj  , jx  oX  ,.jx  noiT  Xa.neXa^ 

fcns  ,i3  Jaanaini  arix  „i  xnioo  aaoe  Xa  naSaX  si  tn.n^^fnt  ,rtx  Ir 

b^OIlO^  nog  8(iy 


=J 


noiaivifcdua  tortXaa  yna  no!  ,.alaixa  xi  tl  Ln  ,i,1x  i,  XiaixVl 


»aaixax.  arfx  lo  rixpnaX  arfx  .vXaXiniiafcni  easaaaoni  n sa  aXnac  x oJnX 


'V 


^ -.flansaxni  nnaaaiR  anx  ^i  V i ylaxlniXatni  Jnisaanaafa 

asaa  aXini.i  a »oai  aaaq  a«  ayfXavlnab  *■  JninilaX  ni  aeiwaJtU  , 

..Xi  50  asnavni  aPx  ai  avixavinat  a,.X  yllanaO  ,aaaa  aXinlXxfi  na  oX 
oxanx  XartX  inxQ  yp  x„o  iaXnioq  aaw  ii' XoX- ..avoXa  ianilat  Xansaxni 
_oviXaviaai  aXX  Xa.iX  Pana  Xnd  saviXavinai  fad  daid.  anoiXonni 


i ’■ 


iild 


-o  Sid  10  sisad.adX  „0  ;.aanas  nnaoaiXi  adx  ni  aXdaa.axni  x\n' ..a 
50  noixinilat  a avas  anvsadaj  .,lano8  1o  snoiXa sixsayn i .^'xa  .b„^ 
■io  a.os  50  anaa  dooX  ina  .,x„,xxa  ni  nataon  - dono  san  doidw  lanyaLi 


lsnoiSo90X3  0B9fiJ 

_,x.dJX  adX  nl  navxi  ylXsuay  ai  aylXa»inal  >o  noiX-iniXai  edl 
' tx  ♦x  i'** 

I i&Il.  " Vd  Vlalaaixodooa  iani.-.axoi  ,d  n«, 


though  not  «ctuaUy  reaching  the  llmitini  case  of  '^-o,,  Such  approx- 
imations are  quite  useful  in  aopZiei  mathematic  a,!  and  for  many  such 
purposes  satisfactory^ 


If  now  we  consider  the  simple  differential  equation 


dy/dx  • f(xyjf  with  the  usual  conditions  on  the  functional  Cauchy 
showed  that  an  lotetjtral  always  exi'sts.j  The  method  generally  used 

for  the  mechanical  integration  of  this  equation  is  an  apDroxiiDation!,i 
which  pushsl  to  the  limiting  ^ase  shows  the  eadstence  of  the  function 
called  the  integrsl!.J  This  can  be  done  graphically  by  strating  from 
a given  point  acoi,i  yo-r^  calculating  by  the  equation  the  slope  of  an 
integral  curve  at  this  point.,1  then  continuing  in  the  corresponding 
direction  for  a short  distance  tc  which  point  a new  slope 

is  calculated*]  This  process  repeated  for  short  enough  intervals 
will  lead  to  a close  approximation  to  the  integral  curve.!  In  symbols 
we  may  Indicate  this  process  as  followsi, 


W(-  ^ 

•«i-4 

from  which  we  have  a set  of  equations  of  the  form 

y{xi)-y(xi^lhf(xi^i,yi^l)(xi- 

If  the  successive  values  are  substituted  in  the  equations 
we  arrive  finally  at  the  value  of  y(xn)  in  terms  of  a sum  of 
products  of  the  form 

*e  may  extend  this  method  to  aoply  also  to  the  oase  of 

systems  of  equations.)  For  instance  let 

iy/dx  - 

dx^'iix  ” •t,i(x)y  * 

Proceeding  to  exoress  the  approximate  results  for  n different  values 
of  *,  and  multiolyini  out  the  results  we  have 

yn®  (l  + .’Zn 


yThe  passage  f ro’n  Zr._i  to  Un  j Zn  is  by  rceans  nf  the  substitution 


**12 

" ■ ... 

# 

,"'To  pass  then  frofu  \io  Zq  to  jt,>,  Zn  woul.1  require  a succession  of  these 
substitutions 

• 3 2 3 1 . 


cr 

-n 


^ We  nay  write  3n  in  the  forno  1+’:^^  where  is  the  matrix  of  the  u's 

! alone^  and  the  product  of  the  succession  of  substitutions  takes  the 
j f orm 

. 1 ■*■  t . vy  ( x • M ) + m '2  ^ yiy  ( ;c  • j f X y j * • • • 

iHhe  summation  bein^  from  1 to  n.  If  we  pas:s  to  the  limit,  we  ar^'ive 


at  inte->'rals  of  a substitution 


1 + f ( x>)  ( ) dx  ^ r r t f Jc  I ) f X 2 ) ( ) ti X idx  2 + * • • 

[See  Volterra:  Leccns  sur  les  functions  des  li.>'ne3^  00  36—42] 


f'  Continuing  our  develoonrent^  let  us  notice  next  the 

(■^equation  d^^^dx  = f(x)u(x),  of  whic'.  we  may  write  the  integral  as  an 
lithe  solution  of  an  inte^'^al  equation.^  namely 
1 h(x)  = + c. 

|lr  a more  general  fo'^m  we  would  have  to  consider  as  linear  integral 
■equations  the  forms 

^ix)  = f{x)-(  x)d^  the  Volterra  equation  of  second  kind^ 

x)=  f Q^(t)F(hjx)  di.  the  Volterra  equation  of  first  Kind^ 


x)  = f(x)~\^ 0 ^>{^-)^'i^,x)dF  the  Preihclm  equation  of  the  seccn  1 kind, 

L '^{x)=\f  (i^^f{t)F(^.x)  dh  the  Fredholm  equation  of  the  first  kind. 

! To  these  we  may  aooly  the  same  metho ds  of  constructiru'  a solution  by 

j the  methods  of  aoproximation  under  discussion^  viz.  those  which 
i consider  the  values  of  the  quantities  at  n ooinl,  in  the  region  under 
u consideration^  passing  ttience  to  the  region  by  lettin.?  n becoiiie 
^'infinite.  An  exaiole  anicnu  tne  eanliest  integral  equations  is 
that  due  to  a problem  of  Abells.  The  pro  b lei;  is  this.  deter  vine 
that  curve  in  a vertical  plane  iowti  which  a xovin.?  00 dy  would  pass 
iin  such  a way  that  it  would  reach  the  lowest  point  in  a time  ^iven 


by  th«  function  whore  '*1  is  the  initial  heij^htui  the  initial 

velocity  bein?  zero.J  If  wOh)  i's  constant  for  all  heiafhts  the 
carve  is  the  tautochrone,\  The  integral  equation  which  has  to  be 
solved  is 

/ x)  " To  ^ 

✓ ( 

Solution;,  uix)*’  ^ 

A dx  /Kx-hJ 

y‘u(x)  is  the  equation  of  the  curve. | 'ISee  Bocherij.t  Linear  Intesfral 
Equationstji  Volterral,'  L'econs  sur  les  equations  inte|rale9.j] 

A generalization  of  this  is  the  eqaati<}n 

f(x)  * f xj  { dB,  where  0<a<l,j 
The  solution  of  this  is 

^ fl^  Abel"s  formula.] 

n dx  vx- “ 

These  equations  treated  by  the  eethod  naentionei  oay  be  found  in 
Volterra:.  Lecona  sur  les  Equations  inte^rales  pp  34  etc.] 


SECOND  LECTURE 

We  return  ^ow  to  the  equation  of  the  second  kind 
The  solution  of  this  equation  is  found  to  be 

'^ix)  * rraf;+/*o^s(Ex;r(a)de 

where  the  function  S is  sfiven  by 


%{x]i)  « -f{xy)4*ixy)^h{xM)*~^^^r^ 

In  these  equations  f is  bounded  and  inferrable. ; The  series  for  3 
is  uniformly  converrent.;  The  meaninr  of  the  ^ over  the  F is  as 

follows:  We  define  for  two  functions  (waiving  for  the  oresent  their 

character) 

which  is  called  oomvoaition  of  the  first  kind.  In  comoositlob  of 
the  second  kind  the  limits  of  integration  are  fixed  j This  ooer^tion 
is  associative^'  but  generally  it  is  not  commutati'^e  ' When  the  ' 
functions  and  P,  permit  of  an  interchanre  without  chanrinr  the 
resultinr  function  they  are  said  to  be  oermutahle.  [See  Colterra^,- 
Fonctions  des  lirnesy  Chapter  IX.;]  If  there  is  riven  a system  of 
oermutable  functions^-  their  compositions  are  also  oermutable  with 
the  oririnal  system  and  with  one  another.'  Also  the  system  arrived 
at  by  addinr  them  or  by  addinr  constant  multiples  of  them  is  also 
permutable.  with  the  others.;  The  rreat  advantare  of  oermutable 
functions  is  thefact  that  the  ordinary  rules  for  handlinr  alrebraic 
rational  expressions  apply  to  them^  and  with  proper  interpretation 
also  the  irrational  processes. 

As  an  instance,  all  the  functions  oermutable  with  T,- 
unity,'  are  of  the  form  F(x~y)  or  F(y-x), 

The  fundamental  theorem  which  bears  upon  expansions  in 

compositions  is  this.;  Let 


OC  OC  <30 

oi  10  i oifi  ^ i 1 1 2*  I* 


a 


I*  t*  t Zfi 


be  a serie**  of  powers  of  the  complex  variables  Zi  Zn  which  is 
convergent  for  values  of  \z^  ^ ffi  Tf'lznl  however  small 

i?i  /?,2,'  -rr!  En  provided  they  are  actual  finite  values,-  then  if  we 
replace  the  variables  z by  functions  Pi^'  , and  place  a s^ar 


— 6- 

‘ov«r  each  F so  that  the  oocration  changes  from  multiplication  of 
variables  to  compo si tioii  of  functions,  the  resulting  expansion  is 
uniformly  conversfent  for  all  values  of  the  variables  x,y  and  for 
any  functions  F which  are  bounded.!  Volterra:.  Fonc  i li^nes  p 140  .f 


This  theorem  included  Borel's  expansion  theorem  as  a special  fonj.; 
p.|  160  loc  cit.j 


Returning  now  to  the  equation  of  the  second  kind 

r(x)  = ro^^(i)F(tx)di 

ie  arrive  at  an  equation  of  the  second  kind,*  and  may  then  solve, 
frequently  by  dif fer- ntl atini  both  sides  as  to  x,  ^ivin^ 

f'(x)  * x)F( x,x)-^f 

It  is  evident  however  that  if  F(x,x)  * 0 we  are  no  better  off  than 
before:,!  as  the  non—homoiieneous  term  drops  out.j  When  this  value  is 
such  that 

F(xtx)  ^ 0 F is  of  order  1.; 


It  is  clear  that  we  have  F(x,y)  * ( x,  y)  where  d(xyj  is  not 


divijiib’le  by  (y^x),  does  not  vanish  for  in  which  case  we  say 

F is  of  order  w.  In  case  ® is  an  integer  it  is  evident  that  m-fold 

differentiation  of  F will  lead  to  anequqtion  of  the  type  desired,! 
so  that  we  can  resolve  the  problem.!  There  are  left  to  consider 

other  types  of  nucleus  of  this  form',  particularly  those  that  have 
algebraic  6r  logarithmic  singularities  for  y"x,  ISee  Lalesco,  Jour.] 
des  Math. I 1908  op  125-2021  .j  In  case  F(x,x)  vanishes  at  a set  of 
point  X which  are  not  a continuum  the  order  becomes  indefinite.' 

Lalesco  has  considered  such  forms,! 

In  the  case  f(x)  * fo  F{tx)  ^dB.  0<a<l 

«e  multiply  both  piles  by  anl  intelrate  fron  0 to  p. 

The  left  sile  is  a Known  function  say  ^(xJ.  The  rilht  side/  by 
interchanging  the  integrals  lakes  the  torn 

f(lx)(y-xJ'‘-Ux-t)-''  dx 

In  deter.ininl  the  new  limits  we  use  a theore*  lue  to  Oirichlet 
In  fact  the  inteiration  is  easily  seen  to  take  olace  over  a 


iri an^l e 


-7- 


havirii?  as  sides  the  u axis^  the  bisector  of  the  an^jle  between  thh'  i? 
xaxisandthej;/axis^anialine^/=a. 

The  expression  F ( e,x)  ( y-x)  ^ evidently  has  the  same 

order  as  F,  We  set  the  last  integral  f - § {B,y)  an  expression  of 
the  first  order^  and  then  have  our  original  equation  reduced  to  one 
of  the  first  kind  which  is  manageable/  [See  Lalesco's  paper  above/J 
We  come  now  to  the  case  with  a logarithmic  singularity: 


f(x)  = To  '^(x)  [\oi,{x-B.)*0\dl  where  C is  ta^en  actually  Euler's 


const  = 0 .•57721--r-, 


1-  The  method  of  recudtion  of  this  equation  actually  presented  is  not 
^ the  one  used  in  the  original  study,  »and  aoioea^s  therefore  to  be 
h highly  artificial.'  We  staBt  with  the  integral 

i:  ! - 

1 r(a)r(e)  ^ r(u+e) 


, This  is  found  easily  by  setting  t~x~(ij-x)z  in  the  form 


rart_ 

ru^t) 


The  left  side  of  the  equation  written  may  be  split  into 
two  parts,  one  containing  u the  other  fc*  only.  We  now  differentiate 
as  to  « and  integrate  as  to  b from  B to  */  The  result  on  the  ri^ht  is 


-iy-x) 


The  result  on  the  left  consists  of  the  two  partstreated  separately,, 
I the  one  containing  u the  other  b/  The  differentiation  as  to  u 
reduces  that  part  to 


'(Fu)'" 

The  other  part  we  write  /\  3 

We  now  take  a=l  3=1,  and  we  finally  have  the  formula 

f ■*'C)  /\(  y,  di  = -(y-i-x) 


Now  if  we  return  to  the  integral  equationn  under  consideration  we  have 


-8- 


Chani^in^  the  variables  we  have 

(E-x)-*-C]/\(j/^)'i^.)  = '-{y-x)  ^ 

From  this 

r o^f(x)/\( yx)dx  =r 0^ /\( yx) dxf o^'viO  [loi?(x-£.)  +Ml  dE, 


~f  o'^4'(^)d^r  yx)  f,lo^(jc-£,)  +CJ  dx 

.The  left  side  therefore  reduces  to  -ro  (^)  ( its  derivative  to 
y is  ‘-f  dtj>  and  second  derivative  as  to  y is  •~^f(y). 


IS 

in 

!'• 

Nr 

'•j 


TEIUD  LECTURE 


t, 


The  process  utilized  to  solve  the  equation  last 
considered  is  an  example  of  a very  general  process  by  which  an  integral 
.equation  with  a ?iven  nucle.us  may  be  reduced  to  anoL.{\er  with  a simpler 
nucleus.;  In  fact  if  we  have  the  equation 

r(xJ  = fo^  si,{OF(lx)dl 

and  multiply  on  both  sides  by  a ^iven  function  ^ (xy)  then  integrate 


as  to  X,  we  have 

(xy)f(x)dx=fo  y§(xy)dxf  o^^>{OF{tx)  dt 
= f o^^fiOdtf  ^^F(E.x)^(  xyJdx 
= /’o^^y(e)  ^{ly)di 

We  may  set  the  result  of  the  composition  Ff=^(xy),  thus  arriving  at 
a new  equation  with  a new  nucleus.'  An  example  is  the  previous  case 
where  the  function  equation  with  nucleus  [ 1 oi  ( jc-£.  ) 

into  one  with  nucleus  -(][/-£.).'  We  now  purpose  to  find  a function  ^ 
which  will  convert  the  nucleus  \oi(  x-i.)*0,  where  C is  the  Kuler 
constant 


We  start  with  the  formula 


r 

' X 


dt 


(y-xj 


a + b—  1 


Tu  re  r(a*e) 

The  meabers  of  this  equation  are  icultipliei  by  • the  parts 

on  the  left  beini  separated  • We  then  repeat  the  process  used  before 
of  differentiation  as  to  u and  integration  as  to  b form  to  . If 


1 

t 


\ 


9- 


T 


we  reoresent  by  P the  expression 

B B pf, 

then  we  have 

r(a+e) 

Nowseta=l  B=landwehavethedesiredresult 

fx^[\oiU-x)-V'l/Tl  +^]P(£f^;cia=-(.v-Jc) 

lln  these  equations  if  we  do  not  take  a=l  we  may  ccaiDlicate  the  nucleus 
lito  a considerable  extent,;  3y  further  differentiation  as  to  a and  in- 
|te?ration  as  to  b it  is  possible  to  arrive  at  expressions  involving 
the  square  of  the  lo^arithiu^  and  even  higher  powers  and  polynorcials 
in  the  lo^.; 

This  leads  us  to  inquire  .what  the  nost  general  integral 
equation  may  be  like.  One  case  studies  by  Lalesco  (Equations  Inte»?rales 
p 127  et  seq)  i s 

lj(xj  = f 

This  is  not  linear^  and  indeed  F may  be  quite  4eneral  Another  general 
form  would  be 

^>(x)=^y(x)+fo  "-F  i{E.x)di.*f  o^'f  0 "-dS.  i,di.  s{e,j.x  ^x)  j.)  yii  s)  ^ 

We  need  not  multiply  these  special  cases  but  pass  at  once  to  the 
consideration  of  the  most  general  case  that  can  occur.;  We  find  necessary 
here  the  general  conception  of  function  of  a line,  and  that  the  most 
ieneral  integral  equation  takes  the  form 

^)(x)^  ^ [yit),  x] 

the  riJht  hand  side  representing  a function  of  all  the  values  4iven 
by  the  function  yU)  where  ^ takes  every  value  from  0 to  1,  and  a 

parameter  ;c,  which  may  also  take  every  value  from  0 to  1.;  This  equation 
may  now  be  studies  as  an  equation  which  is  a limiting  case  of  an 
infinity  of  equations  depending  upon  an  infinity  of  variables.; 

See  Volterra,  Archiv  der  Math/  und  Phys.;  (3)23(1914/5)  Sitzber.; 

Berlin  Math  Ges.;  13(1914)  pp  130-150. ; 


'0^  ■ 

/fr 

U.. 


-10- 


' V « 4 !*• 


Let  u(x)  define  a curve^^  then  any  set  of  numbers  determined  by  some 
process  of  determination  such  that  for  each  function  u(x)  there  is 
one  number, ‘ are  the  function  values  of  a function  of  a line.'  The 
argument  in  this  case  of  the  function  is  the  line  in  a certain  sense, 
or  we  may  say  is  the  totality  of  all  the  function  values  of  yix) 
for  every  vaOue  of  jjc  from  0 to  1.'  A simole  example  is  the  area  under 
the  curve,'  another  is  the  length  of  the  curve.;  In  physics  we  have 
many  such  cases,-  as  for  example  the  ootenytial  at  a ^iven  point  in 
space  due  to  a loop  carrying  an  electric  current  is  a function  of 
the  loop.; 


We  may  study  such  functions  by  our  method  of  development 
depending  on  a set  of  n selected  points  and  the  corresponding 
ordinates,'  so  that  we  would  have  a function  ^ ( Ij  u Us  j ' '' ■' 
Afterwards  we  determine  the  limit  of  the  expressions  as  n approaches 
infinity.;  We  may  also  have  a set  of  such  function 


^ y xj  y 2/  '.Vn ) “ z 1 
^ 9^y X}  y 2 j * i'n ) - z 2 


r • ? • r • ♦ 


• f 


^ y Xf  y 2 j ‘ ‘yn  Zn 

If  we  attach  the  values  z to  the  points  alon^  the  line  we  may  look 
upon  them  as  values  of  a function  z( x) , and  must  then  con  adder  that 
in  place  of  the  index  of  the  F we  must  introduce  jc  as  a parameteif. 


we  have  as  the  limitini?  case 

FlyiOjx]  - z(x) 

• • 

where  the  left  side  is  a function  of  a line  depending  upon  a 
. parameter  x.  This  is  the  most  general  form  of  an  integral  equation 
**  ( of  the  Fredholm  type,-  but  this  includes  all  the  types.) 

^ In  order  to  solve  an  equation  of  this  form  we  expand  it 

||  in  a form  similar  to  a Taylor  Series  ; In  fact  the  exoansic  -i  is  a 
ft  generalization  of  Taylor's  series  from  the  case  of  variables  to 


> that  of  an  infinity  of  variables.'  We  exoand  the  function  a 
1 14. he  in  the  form  . ^ 'J 


^ z(x)=y(  x)  +f  0 o^f  o F git\^  gx)'dCl'iE,.y+"  - 

It  should  be  noticed  that  if  we  stoo  with  the  first  Integral  we'  have 


the  linear  integral  equation.*  A resolvent  function;bis  founds  and 
we  then  have 

yi  x)  = z(x)  + f dti  + f ^zilx^gx)  citidts-^' 

See  Volterra  Fon'  tions  des  li^nes  p.;  69  et  seq.; 


FOURTH  LKCTURE 

Inte^ ro-di f f erenti al  equations  * 

The  equations  contain  the  unknown  function  not  only  unier  the 
integration  si^n  but  also  differentiated  .partially  or  totally  with 

respect  to  the  arguments  it  contains  * As  examples 


df*  dx'-^  to 


which  ^ives  the  motion  of  an  elastic  cor  1 when  hereiitary  effects  are 

considered.,-  that  is  when  the  motion  changes  the  character  of  the 
elasticity  progressively.,*  oroduciriij  an  effect  similar  to  hysteresis 
in  magnetism.’  See  Volterra  Rend.;  Ac  | Real  ; lei  Lincei  191^.,'/ 

Fonctions  des  li^nes  p 97.; 

Another  example  from  physics  is 

t 


£.kiLl  = n(t)*fo\'('^)F('it)d^ 
dt'^ 

to  which  the  solution  of  the  precedinu  is  reduced.; 

From  analysis  we  easily  find  examples.;  We  shall  consider 

here  only  t that  one  arising  in  the  solution-of  the  problem  of 
finding  all  the  functions  oermutable  with  a ^iven  function.'  We 
recall  the  definition  of  permutability 

(xOfO-uidf- 

ks  an  intpolactorv  Raamole  which  will  suiiest  the  general  aethol 
of  attaol^  let  as  take  that  is  we  oropose  to  fini  the  functions 

oermutablewith  v , \ ^ 

^(xy)  = ^(xOdt 

I If  we  differentiate  as  to  k we  have  at  once 

if  Differentiatini  as  to  y we  have  also  3f  31/  i ^ »U 

these  we  eliminate  the  funotioh  and  have  a partial  'i 


-12- 


for  whence 


4/  * F(y-x)  " 

lere  F is  arbitrary  but  differentiable.;  These  functions  f urt  hertnox  e< 
ia1*e  a>Jl  oermutable  with  one  another,'  and  in  this  sense  foriE  a ^rouo. 

his  is  called  for  physical  reasons  the  s^roup  of  closed  cycle, 

(Volterra  Fonctions  des  li^nes  Chap.'VIl),;  The  auestions  as  to  whether 
it  is  true  that  the  iroup  of  functions  peirmutab le  with  3 ^iven  function 
are  therefore  permutable  with  one  another  has  been  answered  affirma- 
tively by  Vessio’t^  C.;R.;  1912  p 682.;  In  this  i^eneral  case  we  make 
use  of  a transformation  upon  the  whole  sis  follows,'  Let  F(xy) 

be  one  of  the  5roup.;  Transform  this  by 

Fi(xy)  * f(  x)F(  xy)<i>(  y) 

and  set  dxi-=dx^r(x)^>(x)  x=K(xi.)  Xj,=  ii(x)  y^^(yi) 

Transform  now  $ ( xy)  in  the  same  way  i?ivin;? 

$x(  xy)=‘f(x)'^(  xy)^(  y) 

Then  we  have  ^ i{xy)=  f ^ x)  dlir  f ^ J r ( x)  F ( xt)^>it)  f it)  Hty) 

^^(  y)dti 

But  from  the  value  of  d^i  we  have  at  once  that  this 

= r ^^f(x)F{xt)^ity)w(y^dt=f(x)Fi^>(y) 

X 1 

That  is  ^^J,(xy)^r(x)^{  xy)^>(  y) 

It  is  evident  no»  that  if  is  the  result  of  a oerautable  oo^nosition 

By  a orooer  choice  of  functions  we  can  reduce  any  function 

P from  the  Jeneral  font  to  a fore  such  that  P(xx)^\,  F xy)  = dP^ ox, 

P.( xu)-dF/Su  and  Fjxx)  =■  0 F,{xx)  '0.;  „ . 

We  now  derive  the  condition  of  pennutability  of  a function 

.tth  a iiven  function  as  the  solution  of  an  intejro-iif ferential 
equation.;  Let  ^ 

dif f erentiaf e as  to  x and  as  to  y 4ivin^ 
iof  which  integral  epuaUcn.- i as  unknown  we  wiU  let  T.  he  the  resolvent. 


■y  1 ' X 

of  which  we  will  let  f,  be  the  resolvent  • 
.Therefore  we  will  have  froo  these  two 


<1. 


X 


A 


♦ 


4 


.Jf, 


Hl3n 

by  subtraction  we  ellfflinate  the  unknown 

I'quatiop'.for  f' 

‘ 


and  have  an 


ae 

nte^ratin?  the  ri^ht  hand  side  by  parts  we  free  the  function  ^ from 
the  differentiation  as  to  and  therefore  have  a froa 

The  solution  given  here  and  the  final  treatment  are  found  in  Foncticns 
de  lignes  pp  61  et  seq«,| 


Let 


FIFTH  lecture; 

Integror-dif ferential  equations.] 

We  found  the  general  form  of  integral  equation  by  passing  to 
the  llsi't  from  a set  of  n equations  to  be 

f1 [y(^)  x] \^z(x) 

1*1 

Similarly  we  arrive  at  an  integrordlf ferential  equationby 
the  consideration  of  a sot  of  simultaneous  dif fer??r44al  equations. 

jy„  z) 

dy^/dz^F giui  j/t  '*t*f*!  Un  z) 

I*  r I* f r I*  r p :•  f i*  r r p '*  r r r r r i*  i*r  r i 

dyn^dz"Fniyi  Vn  Vn  z) 

If  welet  the  number  n increase  now  so  as  to  become  continuously 
infinite  we  find  that  we  must  substitute  for  the  index  a parameter 

X giving  us  x 

dy(  z,  x)^dz  “ F\[y(z  £ 

which  is  an  integrordif ferential  equation.]  We  may  also  arrive  at 
equations  of  this  character  from  partial  differential  equations^!  but 
the  classification  is  not  at  all  easily  settled.]  These  equations 
are  solved  by  the  methods  of  Cauchy  (and  earlier  Laplace)  called 
by  Picard  unethod  of  suocBssiue  aoof'oxifnat  ion  [See  Coursatiji  Hedrick  s 
Trans.]  Vol  2 2 p.j  61  et  seq.jl  iGoursatiji  Vol  III  Chap.]  XXX)  .] 

The  types  of  integrordif ferential  equations  studies  are  classified 
under  three  typcs^i  according  to  the  o^araoterist io  curves  [See 
Bvansj,!  Co'Jloquium  lectures  p 90  et  seq)  i The  types  are  the  elliptic^ 
hyperbolic/  and  parabolic^  The  entire  fidd  is  an  application  of  the 
theory  of  f i o n f functionals  (Evans)  font tionel  1 es 

Frechet),]  functions  in  a Volterra  (that  is  continuously  infinite 


a uiil'  er  ’ o f 
s 11  j I?  e s t s ^ 


iircensions)  spac^,  or  t-he  reporter  of  these  led 
in  brief,,  Volterra  functions. 


Derivative  ani  iifferential 
of  Volterra  functions. 


« 


In  orier  to  pusch  our  frontiers  as  far  bacK  as  possible^  we 
(Eust  consiier  next  the  ilea  of  variable  Volterra  function^  ani  con- 
sepuentl.y  continuity  of  such  function  ani  hence  iifferential  of  such 
function.  A Volterra  function  ieoenclin:?  upon  f{x)  is  continuous 
(Volterra  sense)  if  when  we  substitute  for  fix)  the  function  fix)  + 
where  ^ix)  is  bouniei  in  absolute  value,,  then  the  variation 
of  F\{fix)}\,  that  is  ^ j [ ^ (jc  j f jcj  1 1 -S’|[rf.x:)]|  -<o  is  infinitesitial 

with  t.-  o is  to  be  any  arbitrarily  stcall  quantity.  [There  are 
other  iefinitions  of  continuity  for  a Volterra  function^  for  which 
see  Fonctions  ie  Liiines  o 21.1 

To  iefine  ierivative  at  for-  F we  p’”oc..,ei  as  follows: 


choosing  two  values  of  x,  one  less  than,,  t 
other  greater  than  i,  we  vary  the  function 
fix)  between  these  two  values,  ^eeoin^?  all 
other  values  fixeJ,,  the  variations  of  the 
function  values  bein^  less  in  absolute 
value  than  t.  The  variations  of  F ^ 
are  then  liviiel  by  o the  ar«a  enclose! 
between  the  new  curve  ani  the  original  curve,  ani  the  littit  of  the 
q.uotient  founi.  If  such  limit  exists  it  is  the  ierivative  of  the 
VolterrafunctioriFatE, 

Liffi  = f ' i]j 

It  can  be  usei  to  exo'^ess  the  variation  of  F thus 


tF-^r^  F'  [fix),l.]  I tfix)'il 

wnich  is  entirely  anaios^ous  to  the  formula  for  total  iifferential  of 
a function  of  several  variables,  = ‘ * ‘‘’Fr.  ' . ^e 

^lEay  also  iefine  hij'her  ierivatives,  ani  fini  the  re.markable  tbecrem 
|!that  the  seconi  ierivative  of  a Volterra  function  at  gri  then  is  tn 
■■secQii  ierivative  at  n ani  then  I, 

fit?  may  now  have  as  a Generalization  of  ouc  orece 


-15' 


IQU ^ t i ciB  i'^volvinJ  "ierivgtives  oT  Volterra  f‘uiictioil^3dt..',5;^' 


J * * 

1 id1^  examole^,  analogous  to  the  equation 


• T 


_,,  + • • ♦£naf’''3(/r.  = 0 t'll  *1 

ph^se  solution  is  U 2^  ^-•hn } ' ' ' Un^'^Ur,)-  Constant^  that  is  ^ 

ii  General  any  function  of  zero  orler  ani  homo neo li 'ao  may  have 

as^he  analogous  fortii 

f , x]\  ii(  x)  ix  = 0 

^The  solution  of  this  is  the  Vol terra  function 


p 1 

‘ /’o'-.vCTOiM 

'Ve  may  solve  other  forms  similarly.  A secon.i  examole  is 


y iB  ff’/d 1 + • • • .z/r.  o d yn  = 

vith  solution  + - 'UP  Miere  =’  is  of  ie^ree  ^ero.  The  integral 

equation  »ouli  be  f o h;"  (i)  ss  shovel  A thinq  example  is  ieneralizei 

from  • 

'dP/'dz^ll^F^^y  iy(  y t}  ' ' ' } yrJ  Uy  ~ '- 

which  can  be  inte^rateh  by  first  inteiratin-^  the  sv^tem 

i y 'i* 


dyn  ^ dz-ZuQn 


fhe  Generalized  fOhm  is 

-oP'^dz  ^ r,^p\[y{l),z^\  fo^a(xy)ym)d^dx  - 0 

A soecial  value  ter  P'  ^ives  a soecial  case 

dy(xz)dz)  To '•c(  .xfi)f/(h)'^n. 

1.  -hp  solution  of  the  system  above,  which 

To  solve^this  we  ^ . e J , Solving  these  for 

4.  4 ^ritinu  iovui  any  arbitrary  function  o 

the  n constants  ani  xrtttnj  xon,  y here  we 

results  we  have  the  most  uene’^al  solutio 

have  first  , \ 

y(.x)  ^C(  x)  o^(  x^.\z) 

e +->rtn  Solvinu  the  integral  equation 
where  G is  an  arhitra-,v  function  Golvi  . 

: for  G we  have  ( r \ .r 

Z(x)  ^ y(x)-f  z)y('^.)dc  ^ ^ 

^ n fm-tion  Any  Volterra  function  of  xJ 
where  ^ is  the  resolven  function  ■ .y 

is  the  result  iesi’^ed.  nolications  of  these  theories  to 

l-he-e  are  many  aOO  ^ Volterra  ani  his  students. 

Mechanics.  ?or  *e se  consu 1 t t he  memoi r s 


'‘t.’Kv  i 


Liu  jk  * 


■ i 


-16- 


Latterly  they  have  been  applied  to  the  study  of  Saturn ' sprir^fl*^ 


nd  to  nebulae. 


HISTORICAL.' 

, y The  first  work  done  alon^  these  lines  be^an  with  Volte^ra's 


thesis:  Sopra  alcuni  problemi  della  teoria  del  ootenzial e Pisa  1833 

In  this  infinite  processes  are  considered.'  In  1884  aooeared  an 
* z 

integral  equation  of  the  first  kind\,  ^i>(x)=fo  F ( ix,x)f  ( z,  a)  da,  in 

•Sopra  un  probletua  i el  etro  st  at  ica^ ' Rend.;  Ac.;  Lincei  (3)  6 dp  315—318;^ 

and  in  II  nuove  Cimeuto^  16,  (1834)  9 pp.;  In  1387  the  function  of 

line  (Volterra  function)  aooeared  in  Teoria  delle  equazione 

dif  ferenzial  e lineari,  Soc.;  Ital.‘  d Sci.;  (Xl)  6(1887)  1C7  pp  and 

12  (1899)  69  DO.;  There  are  other  papers  in  the  rend.'  Lincei.' 

In  1896  the  method  of  studying  problems  of  this  type  by  starting 

with  a set  of  al.?ebraic  equations,  whoxe  number  is  then  made 

continuously  infinite,  was  first  used  in  Sulla  inv-.'^^ione  de^li 

inteifrali  defirti,  Atti  Ac..  Torino  31  (1896)  311-523^.  400-408., 

557-567.,  693-708.  See  also  Rend.-  Lincei  (5)  l77-18o^  5*  289-300. 

A summary  of  preceJin:?  results  and  new  results  are  to  be  found  in 

Annali  di  matematica,  (2)  25  (1897)  139-178'  Inte^ro  differential 

equations  were  studies  in  Rend  Lincei  1909.;  Since  then  many  oaoe*s 

have  appeared  Lfp  to  1914^  when  work  was  suspended  for  the  WAR. 


5ti 


■ 'fc- 


r*o  'i  -’• 


.~ir 


'c»S 


^ >■■'  ,■ ' : 


>^-' 


■‘H: 


' ' 'nyJP' 


*>  ^ 

v>  r 


'<y  ,»■{> 


